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I need to draw a prolate cycloid such that it fits a certain length l, and has an integer number of wavelengths. I have these equations for the prolate cycloid:

$$x = h\cos(t+\phi)\cos\theta+at\sin\theta$$ $$y = h(\sin(t+\phi)+1)$$ $$t: 0 ..2\pi n$$ where n is the number of wavelengths.

h here is the amplitude of wave (half the height of the whole figure). a, I believe, is related to where the cutoff point happens in the wave, not quite sure on this one though. I am currently just assigning a to be $\frac{3}{5}h$ which is giving me fairly alright figures. Varying $\theta$ seems to be varying the length of the graph.

Now, how do I set the rotation angle, $\theta$, such that the figure fits in a certain length, l, with a certain number of wavelengths, n?

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We have: $$ x(0)=h\cos(\phi)\cos(\theta) \quad\hbox{and}\quad x(2\pi n)=h\cos(\phi)\cos(\theta)+2\pi na\sin\theta. $$ We want $x(2\pi n)-x(0)=l$, that is $2\pi na\sin\theta=l$, whence: $$ \sin\theta={l\over 2\pi na}. $$

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This worked amazingly! Thank you so much – Youssef Moawad Jul 1 at 15:56

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